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Theorem cnmpt1vsca 17972
Description: Continuity of scalar multiplication; analogue of cnmpt12f 17460 which cannot be used directly because  .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
tlmtrg.f  |-  F  =  (Scalar `  W )
cnmpt1vsca.t  |-  .x.  =  ( .s `  W )
cnmpt1vsca.j  |-  J  =  ( TopOpen `  W )
cnmpt1vsca.k  |-  K  =  ( TopOpen `  F )
cnmpt1vsca.w  |-  ( ph  ->  W  e. TopMod )
cnmpt1vsca.l  |-  ( ph  ->  L  e.  (TopOn `  X ) )
cnmpt1vsca.a  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )
cnmpt1vsca.b  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )
Assertion
Ref Expression
cnmpt1vsca  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Distinct variable groups:    x, F    x, J    x, K    x, L    ph, x    x, W    x, X
Allowed substitution hints:    A( x)    B( x)    .x. ( x)

Proof of Theorem cnmpt1vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . . 7  |-  ( ph  ->  L  e.  (TopOn `  X ) )
2 cnmpt1vsca.w . . . . . . . . 9  |-  ( ph  ->  W  e. TopMod )
3 tlmtrg.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
43tlmscatps 17969 . . . . . . . . 9  |-  ( W  e. TopMod  ->  F  e.  TopSp )
52, 4syl 15 . . . . . . . 8  |-  ( ph  ->  F  e.  TopSp )
6 eqid 2358 . . . . . . . . 9  |-  ( Base `  F )  =  (
Base `  F )
7 cnmpt1vsca.k . . . . . . . . 9  |-  K  =  ( TopOpen `  F )
86, 7istps 16774 . . . . . . . 8  |-  ( F  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  F )
) )
95, 8sylib 188 . . . . . . 7  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  F )
) )
10 cnmpt1vsca.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )
11 cnf2 17079 . . . . . . 7  |-  ( ( L  e.  (TopOn `  X )  /\  K  e.  (TopOn `  ( Base `  F ) )  /\  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F ) )
121, 9, 10, 11syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F )
)
13 eqid 2358 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1413fmpt 5761 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  F
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  F
) )
1512, 14sylibr 203 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  F ) )
1615r19.21bi 2717 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  F
) )
17 tlmtps 17966 . . . . . . . . 9  |-  ( W  e. TopMod  ->  W  e.  TopSp )
182, 17syl 15 . . . . . . . 8  |-  ( ph  ->  W  e.  TopSp )
19 eqid 2358 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
20 cnmpt1vsca.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
2119, 20istps 16774 . . . . . . . 8  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  ( Base `  W )
) )
2218, 21sylib 188 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  W )
) )
23 cnmpt1vsca.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )
24 cnf2 17079 . . . . . . 7  |-  ( ( L  e.  (TopOn `  X )  /\  J  e.  (TopOn `  ( Base `  W ) )  /\  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W ) )
251, 22, 23, 24syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W )
)
26 eqid 2358 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
2726fmpt 5761 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  W
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  W
) )
2825, 27sylibr 203 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  W ) )
2928r19.21bi 2717 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  W
) )
30 eqid 2358 . . . . 5  |-  ( .s f `  W )  =  ( .s f `  W )
31 cnmpt1vsca.t . . . . 5  |-  .x.  =  ( .s `  W )
3219, 3, 6, 30, 31scafval 15739 . . . 4  |-  ( ( A  e.  ( Base `  F )  /\  B  e.  ( Base `  W
) )  ->  ( A ( .s f `  W ) B )  =  ( A  .x.  B ) )
3316, 29, 32syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( .s f `  W ) B )  =  ( A  .x.  B ) )
3433mpteq2dva 4185 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .s f `  W ) B ) )  =  ( x  e.  X  |->  ( A  .x.  B
) ) )
3530, 20, 3, 7vscacn 17964 . . . 4  |-  ( W  e. TopMod  ->  ( .s f `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
362, 35syl 15 . . 3  |-  ( ph  ->  ( .s f `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
371, 10, 23, 36cnmpt12f 17460 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .s f `  W ) B ) )  e.  ( L  Cn  J
) )
3834, 37eqeltrrd 2433 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619    e. cmpt 4156   -->wf 5330   ` cfv 5334  (class class class)co 5942   Basecbs 13239  Scalarcsca 13302   .scvsca 13303   TopOpenctopn 13419   .s fcscaf 15721  TopOnctopon 16732   TopSpctps 16734    Cn ccn 17054    tX ctx 17355  TopModctlm 17936
This theorem is referenced by:  tlmtgp  17974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-map 6859  df-slot 13243  df-base 13244  df-topgen 13437  df-scaf 15723  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cn 17057  df-tx 17357  df-tmd 17851  df-tgp 17852  df-trg 17938  df-tlm 17940
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